## Abstract

Let ℓ be any fixed prime number. We define the ℓ-Genocchi numbers by G_{n}:=ℓ(1−ℓ^{n})B_{n}, with B_{n} the n-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime p is ℓ-Genocchi irregular if it divides at least one of the ℓ-Genocchi numbers G_{2},G_{4},…,G_{p−3}, and ℓ-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of ℓ-Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ=2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1−ℓ^{n})B_{2n}/2n and (1+ℓ^{n})B_{2n}/2n. This allows us to estimate the number of primes p≤x for which there exist modulo p Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level ℓ.

Original language | English |
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Pages (from-to) | 147-184 |

Number of pages | 38 |

Journal | Journal of Number Theory |

Volume | 251 |

DOIs | |

Publication status | Published - Oct 2023 |

Externally published | Yes |

## Keywords

- Artin's primitive root conjecture
- Ramanujan type congruences
- ℓ-Genocchi numbers
- ℓ-regularity